Robert G. Melton Department of Aerospace Engineering Penn State University 6 th International Workshop and Advanced School, “Spaceflight Dynamics and Control”

Robert G. MeltonDepartment of Aerospace EngineeringPenn State University

6th International Workshop and Advanced School, “Spaceflight Dynamics and Control” Covilhã, Portugal March 28-30, 2011

Numerical analysis of constrained time-optimal satellite reorientation

Gamma-Ray Bursts/ Swift• First detected by Vela satellites in 1960’s• Source: formation of black holes or neutron star collsions• Intense gamma-ray burst, with rapidly fadiing afterglow (discovered by Beppo-SAX satellite)

• Swift detects burst with wide-FOV detector, then slews to align narrow-FOV telescopes (X-ray, UV/optical)

– Sensor axis must avoid Sun, Earth, Moon (“keep-out” zones – constraint cones)

2

Unconstrained Time-Optimal Reorientation

• Bilimoria and Wie (1993) unconstrained solution NOT eigenaxis rotation• spherically symmetric mass distribution• independently and equally limited control torques• bang-bang solution, switching is function of reorientation angle

• Others examined different mass symmetries, control architectures

• Bai and Junkins (2009) • discovered different switching structure, local optima• for magnitude-limited torque vector, solution IS eigenaxis rotation

3

Constrained Problem (multiple cones): No Boundary Arcs or Points Observed

Example:0.1 deg. gap between Sun and Moon cones

4

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1

2Angular Velocity

time

i

1

2

3

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1

2Euler Parameters

time

i

1

2

3

4

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1Control Torques

time

Mi

M1

M2

M3

tf = 3.0659, 300 nodes, 8 switches

Constrained Problem (multiple cones)

5

AA ˆˆcos 1

A

A

Keep-out Cone Constraint

(cone axis for source A)

(sensor axis)

6

4321

321

4321

321

H

ftJ min

0ˆˆcos 1 AAC

CHH ~

00

00

C

C

if

if

0C :conditiontangency plus

Optimal Control Formulation

Resulting necessary conditions are analytically intractable

7

maxmaxmax

max

M

MM

I

II

M

I ii

iiii

Numerical Studies

1. Sensor axis constrained to follow the cone boundary (forced boundary arc)

2. Sensor axis constrained not to enter the cone3. Entire s/c executes -rotation about A

Legendre pseudospectral method used(DIDO software)

Scaling:

fi ˆ,ˆ lie on constraint cone

• I1 = I2 = I3 and M1,max = M2,max = M3,max • lies along principal body axis b1

• final orientation of b2, b3 generally unconstrained8

Case BA-1 (forced boundary arc)

• A = 45 deg. (approx. the Sun cone for Swift)• Sensor axis always lies on boundary• Transverse body axes are free• = 90 deg.

A

A

i

f

9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

Mi

M1

M2

M3

Case BA-1 (forced boundary arc)

tf = 1.9480, 151 nodes 10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.04

-1.02

-1

-0.98

-0.96

time

Ha

milt

on

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1500

-1000

-500

0

500

time

cost

ate

s

1

2

3

1

2

3

4

Case BA-1 (forced boundary arc)

11

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

time

Mi

M1

M2

M3

Case BA-2 (forced boundary arc)

• A = 23 deg. (approx. the Moon cone for Swift)• Sensor axis always lies on boundary• Transverse body axes are free• = 70 deg.

tf = 1.3020, 100 nodes 12

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.04

-1.02

-1

-0.98

-0.96

time

Ha

milt

on

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4-300

-200

-100

0

100

time

cost

ate

s

1

2

3

1

2

3

4

Case BA-2 (forced boundary arc)

13

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

dis

tan

ce fr

om

"ke

ep

-ou

t" c

on

e (

rad

)

time

Case BP-1 • same geometry as BA-1 (A = 45 deg., = 90 deg.)• forced boundary points at initial and final times• sensor axis departs from constraint cone

tf = 1.9258 (1% faster than BA-1)

250 nodes

Angle between sensor axis and constraint cone

14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

Mi

M1

M2

M3

Case BP-1

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.08

-1.06

-1.04

-1.02

-1

-0.98

time

Ha

milt

on

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3000

-2000

-1000

0

1000

2000

time

cost

ate

s

1

2

3

1

2

3

4

Case BP-1

16

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.002

0.004

0.006

0.008

0.01

0.012

dis

tan

ce fr

om

"ke

ep

-ou

t" c

on

e (

rad

)

time

• same geometry as BA-2 (A = 23 deg., = 70 deg.)• forced boundary points at initial and final times• sensor axis departs from constraint cone

Case BP-2

tf = 1.2967 (0.4% faster than BA-2)

100 nodes

Angle between sensor axis and constraint cone

17

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

time

Mi

M1

M2

M3

Case BP-2

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.1

-1.05

-1

-0.95

-0.9

time

Ha

milt

on

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4-15000

-10000

-5000

0

5000

time

cost

ate

s

1

2

3

1

2

3

4

Case BP-2

19

Sensor axis path along the constraint boundary

i

A ˆˆ

f

Constrained Rotation Axis

Entire s/c executes -rotation • sensor axis on cone boundary• rotation axis along cone axis

20

maxmax /ˆ MM

),,max( 321max

M

It f 2,

Problem now becomes one-dimensional, with bang-bang solution

Applying to geometry of:

BA-1 tf = 2.1078 (8% longer than BA-1)

BA-2 tf = 2.0966 (37% longer than BA-2)

Constrained Rotation Axis

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Practical Consideration

• Pseudospectral code requires 20 minutes < t < 12 hours (if no initial guess provided)• Present research involves use of two-stage solution:

1. approx soln S (via particle swarm optimizer)2. S = initial guess for pseudospectral code

(states, controls, node times at CGL points)

Successfully applied to 1-D slew maneuver

22

23

0 0.5 1 1.5 2 2.5-2

-1.5

-1

-0.5

0

0.5

1

1.51-D Slew Maneuver

time

con

tro

l to

rqu

e

PSO valuesChebyshev approx

0 0.5 1 1.5 2 2.5 3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Dido Result -- 200 nodes, accurate mode

time

con

tro

l to

rqu

e

Dido

No guesscpu time = 148 sec.

With PSO guesscpu time = 76 sec,

Conclusions and Recommendations• For independently limited control torques, and

initial and final sensor directions on the boundary:• trajectory immediately departs the boundary • no interior BP’s or BA’s observed• forced boundary arc yields suboptimal time

• Need to conduct more accurate numerical studies• Bellman PS method• Interior boundary points? (indirect method)

• Study magnitude-limited control torque case• Implementation

• expand PSO+Dido to 3-D case

24

fin

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